Could it be mile-a-minute vine? That's sort of thorny and grows really fast. You might try pinging that name and see what comes up. If that is what it is, get rid of it as quickly as possible, and do not let it flower and seed. It is one of the worst invaders in the past 20 years, and spreads fast.

What about Round-up? Or you could try digging at its base to excavate the roots, since it seems to have some energy stored down there. Sometimes if you keep pulling it, it will run out of energy and give up.

wardw: I looked at some images of mile.a.minute, I don't think it is..
if you notice the leaves are more oval/rounded on this vine..
the mile-a-minute seems to have a triangular look to the leaf.
Unless it changes as it gets older, I dunno, I try to get it every week or two.

ladychroe: I use a vine killer that says it does blackberry, kudzu, and poison ivy.. it seems to kill it,
but I don't think its one main plant...each single vine is its own entity.. i kill one, and before
I blink, another one is trying to climb 2-3 ft up the fence! 20 ft away..
It's just too hard to dig where its growing, there is a retaining wall right behind it..and no room to move,
I reach/lean over the fence to grab the vinelings/spray, and to take those pics..

I have gotten rid of this horrible weed, I think. I dug down about 12", and dug out the tuber. Some were the size of a potatoe. Most I had to dig twice to finally eradicate them, meaing I haven't seen any in 2 years. If they are coming from your neighbor, I would say you have an never-going-to-win battle, unless you can get permission to remove it from your neighbor's yard. Some people have had luck with placing a cut vine into undiluted roundup, and leaving for a period of time. The idea is to have the vine suck up the poison directly into the roots. I haven't had to resort to that, so I don't know all the particulars.

This is a case where a picture would have been worth a thousand words. We rather blessed with a few smilax species in New Jersey, and I must admit I like "sticker vine" as we used to call it. Birds eat the berries and it comes up where ever they perch.